Theorem unipr 2505

Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16.

Hypotheses

Ref	Expression
unipr.1	|- A e. V
unipr.2	|- B e. V

Assertion

Ref	Expression
unipr	|- U.{A, B} = (A u. B)

Proof of Theorem unipr

Step	Hyp	Ref	Expression
1		19.43 1084	. . . 4 |- (E.y((x e. y /\ y = A) \/ (x e. y /\ y = B)) <-> (E.y(x e. y /\ y = A) \/ E.y(x e. y /\ y = B)))
2		visset 1804	. . . . . . . 8 |- y e. V
3	2	elpr 2414	. . . . . . 7 |- (y e. {A, B} <-> (y = A \/ y = B))
4	3	anbi2i 479	. . . . . 6 |- ((x e. y /\ y e. {A, B}) <-> (x e. y /\ (y = A \/ y = B)))
5		andi 602	. . . . . 6 |- ((x e. y /\ (y = A \/ y = B)) <-> ((x e. y /\ y = A) \/ (x e. y /\ y = B)))
6	4, 5	bitr 173	. . . . 5 |- ((x e. y /\ y e. {A, B}) <-> ((x e. y /\ y = A) \/ (x e. y /\ y = B)))
7	6	exbii 1047	. . . 4 |- (E.y(x e. y /\ y e. {A, B}) <-> E.y((x e. y /\ y = A) \/ (x e. y /\ y = B)))
8		unipr.1	. . . . . . 7 |- A e. V
9	8	clel3 1884	. . . . . 6 |- (x e. A <-> E.y(y = A /\ x e. y))
10		exancom 1050	. . . . . 6 |- (E.y(y = A /\ x e. y) <-> E.y(x e. y /\ y = A))
11	9, 10	bitr 173	. . . . 5 |- (x e. A <-> E.y(x e. y /\ y = A))
12		unipr.2	. . . . . . 7 |- B e. V
13	12	clel3 1884	. . . . . 6 |- (x e. B <-> E.y(y = B /\ x e. y))
14		exancom 1050	. . . . . 6 |- (E.y(y = B /\ x e. y) <-> E.y(x e. y /\ y = B))
15	13, 14	bitr 173	. . . . 5 |- (x e. B <-> E.y(x e. y /\ y = B))
16	11, 15	orbi12i 257	. . . 4 |- ((x e. A \/ x e. B) <-> (E.y(x e. y /\ y = A) \/ E.y(x e. y /\ y = B)))
17	1, 7, 16	3bitr4r 184	. . 3 |- ((x e. A \/ x e. B) <-> E.y(x e. y /\ y e. {A, B}))
18	17	abbii 1567	. 2 |- {x | (x e. A \/ x e. B)} = {x | E.y(x e. y /\ y e. {A, B})}
19		df-un 2040	. 2 |- (A u. B) = {x | (x e. A \/ x e. B)}
20		df-uni 2494	. 2 |- U.{A, B} = {x | E.y(x e. y /\ y e. {A, B})}
21	18, 19, 20	3eqtr4r 1498	1 |- U.{A, B} = (A u. B)

Syntax hints:   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  Vcvv 1802   u. cun 2035  {cpr 2400  U.cuni 2493

This theorem is referenced by:  uniprg 2506  unisn 2507  uniop 2797  unex 2863  rankxplim 4684  indistop 7590  indistps 7595  dfchj3 9240  mapudiscn 10399

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-uni 2494

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